New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation (Q864595)

The authors study the existence and uniqueness of positive solutions of the operator equation \[ A(x,x)+Bx=x,\quad x\in E, \] where \(E\) is a Banach space ordered by a normal cone, \(A\) is a mixed monotone operator with convexity and concavity, and \(B\) is an affine operator. Some applications to the nonlinear integral equation \[ \int_{\mathbb{R}^n} K(t,s)[f(x(s))+g(x(s))]\,ds= [1+G_1(t)]x(t)-G_2(t)x(t+\tau)-G_3(t), \] where \(t\), \(\tau\in \mathbb{R}^n\), are also given.